Problem 71 - Algebra 2

Question

What is the simplified form of \(\frac{3-\sqrt{5}}{\sqrt{5}} ?\) $$ \begin{array}{ll}{\text { A. } \frac{3 \sqrt{5}-5}{5}} & {\text { B. } \frac{5 \sqrt{3}-5}{5}} \\ {\text { C. } \sqrt{3}-\sqrt{15}} & {\text { D. } \frac{14-6 \sqrt{5}}{5}}\end{array} $$

Step-by-Step Solution

Answer

The simplified form of \(\frac{3-\sqrt{5}}{\sqrt{5}}\) is \(\frac{3\sqrt{5}-5}{5}\), which matches Answer Choice A.

Step 1: Identify the Conjugate

The conjugate of any binomial \(a + b\) is the binomial with the opposite sign in the middle: \(a - b\). Here, since we only have \(\sqrt{5}\) in our denominator, we consider this as \(0 + \sqrt{5}\). Therefore, the conjugate is \(0 - \sqrt{5}\), or \(-\sqrt{5}\).

Step 2: Multiply by the Conjugate

Multiply the numerator and denominator by the conjugate, \(-\sqrt{5}\). This gives:\[\frac{3-\sqrt{5}}{\sqrt{5}} \times \frac{-\sqrt{5}}{-\sqrt{5}} = \frac{-3\sqrt{5}+5}{-5}\]

Step 3: Simplify

Simplify the resulting expression:\[\frac{-3\sqrt{5}+5}{-5} = -\frac{3\sqrt{5}-5}{5}\]But since we generally prefer to avoid negative fractions, we multiply top and botton by -1:\[-\frac{3\sqrt{5}-5}{5} = \frac{3\sqrt{5}-5}{5}\]This is our final simplified form.

Key Concepts

Conjugate Pairs

Conjugate pairs are a helpful concept in algebra, especially when dealing with binomials and radicals in denominators. When you have a binomial such as \(a + b\), its conjugate will be \(a - b\). This means you simply change the sign in the middle. Conjugates are often used to rationalize expressions, especially those with square roots in the denominator.

Using conjugates helps eliminate radicals in the denominator. When you multiply a binomial by its conjugate, you apply the difference of squares formula: \((a + b)(a - b) = a^2 - b^2\). This operation removes the radical because the result is a simple expression with no square root, allowing for further simplification.

In our exercise, \(rac{3-\sqrt{5}}{\sqrt{5}}\), we consider \(\sqrt{5}\) as \(0 + \sqrt{5}\) and identify its conjugate as \(-\sqrt{5}\). This multiplication helps turn a difficult expression into a more manageable algebraic fraction.

Simplifying Radical Expressions

Simplifying radical expressions involves reducing them into their simplest form, sometimes involving rationalization or other algebraic manipulations. Radical expressions have square roots, cube roots, or other roots, and simplifying them usually means ensuring they do not have any radicals in the denominator.

In our problem, we had \(\frac{3-\sqrt{5}}{\sqrt{5}}\), where \(\sqrt{5}\) was in the denominator. To simplify this, you multiply both the numerator and the denominator by the conjugate of the denominator, \(-\sqrt{5}\).

• After multiplying, you get \(-\frac{3 \sqrt{5} - 5}{5}\), leaving no radicals in the denominator.
• The expression further simplifies itself to \(\frac{3 \sqrt{5} - 5}{5}\), presenting it in the simplest form.

This process ensures the original expression is equivalent but free from complicated radical terms, especially in the denominator, making it easier to work with in subsequent algebraic computations.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, the denominator, or both include algebraic expressions containing variables or radicals. The main goal when working with algebraic fractions is to simplify them into their simplest form, which often involves operations like factoring, canceling common terms, and rationalizing denominators.

In dealing with the fraction \(\frac{3-\sqrt{5}}{\sqrt{5}}\), we needed to simplify the expression by rationalizing the denominator using a conjugate pair. Once this simplification is applied, the algebraic fraction becomes easier to interpret and utilize.

• We started with an expression that had a radical in the denominator, which often complicates further algebraic manipulations.
• Through multiplication by the conjugate, we change the form of the expression, resulting in an equivalent but more manageable algebraic fraction: \(\frac{3 \sqrt{5} - 5}{5}\).

Understanding and simplifying algebraic fractions are crucial skills, especially for solving algebra problems where expressions need to be combined, compared, or computed together.